The optimization process and recommendations:

1. Avoid Selecting Unnecessary Columns (query line: 2): Avoid selecting all columns with the '*' wildcard, unless you intend to use them all. Selecting redundant columns may result in unnecessary performance degradation.
2. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
3. Replace In Subquery With Correlated Exists (modified query below): In many cases, an EXISTS subquery with a correlated condition will perform better than a non correlated IN subquery.
4. Use UNION ALL instead of UNION (query line: 49): Always use UNION ALL unless you need to eliminate duplicate records. By using UNION ALL, you'll avoid the expensive distinct operation the database applies when using a UNION clause.

Optimal indexes for this query:

ALTER TABLE `accounts` ADD INDEX `accounts_idx_type_account_id` (`type`,`account_type_id`);
ALTER TABLE `accounts` ADD INDEX `accounts_idx_account_id_type` (`account_type_id`,`type`);

The optimized query:

SELECT
        * 
    FROM
        accounts 
    WHERE
        NOT EXISTS (
            SELECT
                1 
            FROM
                ((SELECT
                    accounts.id 
                FROM
                    accounts 
                WHERE
                    accounts.account_type_id IN (
                        1, 2
                    ) 
                    AND accounts.`type` = 'single' 
                ORDER BY
                    RAND() LIMIT 5) 
            UNION
            (
                SELECT
                    accounts.id 
                FROM
                    accounts 
                WHERE
                    accounts.account_type_id IN (
                        1, 2
                    ) 
                    AND accounts.`type` = 'joint' 
                ORDER BY
                    RAND() LIMIT 5
            ) 
        UNION
        (
            SELECT
                accounts.id 
            FROM
                accounts 
            WHERE
                accounts.account_type_id = 3 
                AND accounts.`type` = 'single' 
            ORDER BY
                RAND() LIMIT 5
        ) 
    UNION
    (
        SELECT
            accounts.id 
        FROM
            accounts 
        WHERE
            accounts.account_type_id = 3 
            AND accounts.`type` = 'joint' 
        ORDER BY
            RAND() LIMIT 5
    )
) a 
WHERE
(
    accounts.id = a.id
)
)